OK, so I proved that my two line integrals along C1 and C2 are equal.
我证出了分别沿着C1和C2的两个线积分是相等的。
We've learned about double integrals, and we've learned about line integrals.
我们已经学过，二重积分和线积分。
It is not a surprise that you will get the same answer for both line integrals.
对它们两个做线积分，得到相同结果，就不令人感到吃惊了。
Let me just switch gears completely and switch to today's topic, which is line integrals and work in 3D.
我们换一个话题,开始今天的内容，线积分和3维空间中的“功”
You have three line integrals to compute instead of two, but conceptually it remains exactly the same idea.
要计算三个线积分，而不是两个，但概念上是一样的。
That is the key to computing things in practice. It means, actually, you already know how to compute line integrals for flux.
这是计算过程的关键步骤，也就是说，你们已经知道如何用线积分去计算通量了。
So, the fundamental theorem of calculus, not for line integrals, tells you if you integrate a derivative, then you get back the function.
微积分基本定理，不是曲线积分的，告诉我们，如果对函数的导数积分，就会得回原函数。
And whether these line integrals or double integrals are representing work, flux, integral of a curve, whatever, the way that we actually compute them is the same.
不管是线积分或是二重积分，也不管它们表示的是功还是通量，计算它们的方法实际上是一样的。
When I sum all of the little line integrals together, all of the inner things cancel out, and the only ones that I go through only once are those that are at the outer most edges.
当将所有线积分加在一起时，所有内部的线积分都抵消了，只剩下仅需要一次计算的，就是外围边缘部分。
This paper makes some supplement for the error in demonstrating the relations between two curve line integrals in the textbook advanced mathematics (edited by Tongji university 4th edition).
对教材《高等数学》（同济大学数学教研室主编，第四版）中关于两类平面曲线积分关系证明中的一处疏漏给出补充证明。
This paper introduces the concept of potential function into the multiply-connected region and provides its method of computation in order to solve some complex problems concerning line integrals.
本文在复连域上引进势函数的概念，并给出其计算方法，以此来解决一些复杂的曲线积分计算问题。
Line integrals are used extensively in the theory of functions of a complex variable.
在复变量函数理论中广泛使用线积分。
Line Integrals, Potentials, Curl and Gradient in 3d (Omit: Electricity and magnetism).
三维空间的线积分，位势，旋度和梯度(省略：电磁场)。
Line Integrals, Potentials, Curl and Gradient in 3d (Omit: Electricity and magnetism).
三维空间的线积分，位势，旋度和梯度(省略：电磁场)。
应用推荐